A Two Weight Inequality for the Hilbert transform Assuming an Energy Hypothesis

TL;DR

This paper characterizes the two weight inequality for the Hilbert transform under certain side conditions, introducing an Energy Condition that captures weight distribution and relates to existing pivotal conditions.

Contribution

It introduces a new Energy Condition that, combined with known testing conditions, characterizes the two weight inequality for the Hilbert transform under energy hypotheses.

Findings

Energy Condition is a consequence of the three main conditions.

Counterexample shows Pivotal Conditions are not necessary.

Energy Hypotheses are necessary at one endpoint for the inequality.

Abstract

Subject to a range of side conditions, the two weight inequality for the Hilbert transform is characterized in terms of (1) a Poisson A_2 condition on the weights (2) A forward testing condition, in which the two weight inequality is tested on intervals (3) and a backwards testing condition, dual to (2). A critical new concept in the proof is an Energy Condition, which incorporates information about the distribution of the weights in question inside intervals. This condition is a consequence of the three conditions above. The Side Conditions are termed 'Energy Hypotheses'. At one endpoint they are necessary for the two weight inequality, and at the other, they are the Pivotal Conditions of Nazarov-Treil-Volberg. This new concept is combined with a known proof strategy devised by Nazarov-Treil-Volberg. A counterexample shows that the Pivotal Condition are not necessary for the two weight inequality.